1
Estimation of unsteady aerodynamics in the wake of a freely flying European starling
Hadar Ben-Gida1, Adam Kirchhefer2, Zachary J. Taylor1, Wayne Bezner-Kerr3, Christopher G. Guglielmo3, Gregory A. Kopp2 and Roi Gurka4
1 School of Mechanical Engineering, Tel-Aviv University, Tel-Aviv, 69978, Israel 2 Boundary Layer Wind Tunnel Laboratory, Faculty of Engineering, University of Western Ontario, London, N6A 5B9, Canada 3 Department of Biology, Advanced Facility for Avian Research, University of Western Ontario, London, N6A 5B9, Canada 4 Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel
Abstract
Wing flapping is one of the most widespread propulsion methods found in nature;
however, the current understanding of the aerodynamics in bird wakes is incomplete. The role
of the unsteady motion in the flow and its contribution to the aerodynamics is still an open
question. In the current study, the wake of a freely flying European starling has been
investigated using long-duration high-speed Particle Image Velocimetry (PIV) in the near
wake. Kinematic analysis of the wings and body of the bird has been performed using
additional high-speed cameras that recorded the bird movement simultaneously with the PIV
measurements. The wake evolution of four complete wingbeats has been characterized
through reconstruction of the time-resolved data, and the aerodynamics in the wake have been
analyzed in terms of the streamwise forces acting on the bird. The profile drag from classical
aerodynamics was found to be positive during most of the wingbeat cycle, yet kinematic
images show that the bird does not decelerate. It is shown that unsteady aerodynamics are
necessary to satisfy the drag/thrust balance by approximating the unsteady drag term. These
findings may shed light on the flight efficiency of birds by providing a partial answer to how
they minimize drag during flapping flight.
1 Introduction
Flapping flight is one of the most complex yet widespread propulsion methods found in
nature. Although aeronautical technology has advanced remarkably over the past century,
flying animals still demonstrate higher efficiency. One of the key open questions is the role of
unsteady fluid motion in the wake of flying animals, and its contribution to the forces acting
during the downstroke and upstroke [1]. The unsteady flow over small-scale wings has gained
2
significant attention recently, both in the study of bird and insect flight, as well as to develop
advanced aerodynamic models for high-performance micro-aerial vehicles [2]). The goal of
the current study is to examine both the steady and the unsteady aerodynamics in the wake of
a freely flying bird with a particular focus on the propulsive forces.
Among the first few attempts to describe unsteady aerodynamics [3-5], Brown [5]
distinguished several patterns of flapping flight, and described complex movements of the
wings through multiple sets of illustrations for different type of birds. Recently, Brunton and
Rowley [6] developed reduced-order models for the unsteady aerodynamic forces on a small
wing in response to agile maneuvers and gusts based on the framework suggested by Wagner
[3] and Theodorsen [4]. However, they did not manage to augment their model with non-
linear stall and separation models, which are important for improving lift modeling (see for
example: Henningsson et al. [7]).
According to quasi-steady-state aerodynamic theory, slow-flying vertebrates should not
be able to generate enough lift to remain aloft [8]. Therefore, unsteady aerodynamic
mechanisms to enhance lift production have been proposed. Muijres et al. [9] showed that
unsteady aerodynamic mechanisms are used not only by insects but also by larger and heavier
fliers. Hubel and Tropea [10] verified Muijres et al.’s [9] findings by showing that the
unsteady effects are not negligible for a goose-sized flapping model. Thus far, the main
purpose of investigating unsteady aerodynamic mechanisms has been to understand their
ability to enhance lift generation. However, it remains relatively unknown how unsteady
aerodynamics participate in the drag and thrust balance of flapping flight.
The complex unsteady features of flapping flight introduce challenges to any realistic
aerodynamic analysis. One of the first attempts to incorporate realistic wake structure in an
aerodynamic model of bird flight was by Rayner [11-13] who proposed that each wingbeat
was only aerodynamically active during the downstroke. As part of the wake structure,
starting and ending vortices were suggested to be produced at the beginning and at the end of
the downstroke phase, respectively. These vortices are connected by a pair of trailing vortices
shed from the wingtips [11]. Therefore, at a certain distance downstream, the wake was
assumed to be composed of a series of vortex rings referred to as ‘elliptical loops’ [12] which
were conceptually related to the bird’s wingspan and the circulation dictated by the force
requirements (lift, profile drag and parasitic drag) of the bird. However, Rayner's model did
not match later experimental observations. Spedding [14] performed measurements of vortex
circulation in a jackdaw (Corvus monedula) wake and indicated that approximately half of the
required momentum for weight support was present for jackdaw flight. As a consequence, the
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‘wake momentum paradox’ arose. It was concluded that the discrepancy could be a result of
unidentified complexities in the wake structure; i.e., Rayner's model was too idealized.
Spedding [15] performed experiments, with the same apparatus as for the jackdaw on kestrel
(Falco tinnunculus) flight at moderate (U∞=7 m/s) speeds and observed a distinctly different
wake topology. He suggested that instead of discrete loops separated by aerodynamically
inactive upstrokes, two continuous undulating vortex tubes were found in the wake; i.e., the
upstrokes were also aerodynamically active. The measured circulation of the shed vortices
was similar during the downstroke and upstroke, and it was adequate for supporting the
weight of the bird. In addition to the lift force, it was found that generating net thrust occurred
through varying the wing geometry, not through varying the circulation, implying that the
wing motion is important.
The concept of different wake topologies has prompted analyses of vortex gaits [16-18].
Rayner [16] characterized the vortex gait selection for flying birds and stated that the choice
between the different vortex gaits is determined by flight speed and wing morphology.
Spedding et al. [18] investigated the wake structure behind a thrush nightingale (Luscinia
luscinia). They concluded that the structure of the wake downstream, far from the body
(roughly 17 chord lengths), varies gradually from an approximately elliptical vortex wake to a
continuous trailing vortex wake. Therefore, the wake is comprised neither of a series of
elliptical vortex loops, nor a pair of continuous trailing vortices, but is a combination of both.
Eventually, the ‘wake momentum paradox’ was addressed, within the bounds of experimental
uncertainty, through the high spatial resolution available in PIV, as well as a detailed
accounting procedure for the calculation of circulation in the wake [18]. However, the
relationship between specific wake topology and the propulsive aerodynamics of bird flight
remains an open question.
The vorticity structure in the wake of a flying bird, similar to the distribution of
vorticity in any wake, is dependent on the boundary conditions. Consequently, the kinematics
of the wings has been investigated in the literature with the goal of either using the wing
motion to predict forces or associating the wake topology with the motion of the wings. Two
wingbeat kinematics of a thrush nightingale [19], and of two individual robins [20], were
quantified in order to relate them to their vortex wakes. However, the kinematic variations
with flight speed occurred only during the upstroke period where the wing folding and the
wingbeat frequency were observed to vary. In addition to the wingbeat kinematics, the
streamwise distance between the bird and the measurement location has been shown to be of
importance when drawing conclusions about the aerodynamics related to the wake structure.
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Hedenström et al. [20] investigated wakes behind European robins (Erithacus rubecula) and
found that they resemble the thrush nightingale wake [18]. It was argued that the wakes'
circulations were similar because the measurements were in the far wake, significantly
downstream of the bird [1]. Studying both the near and far wake of Pallas' long tongued bat
(Glossophaga soricina), Johansson et al. [21] concluded that measurements in the far wake
might lead to misinterpretation of the wake topology. This misinterpretation occurs because
the near wake is more readily tied to the generating wing kinematics and, thus, contains
details of vortex structures that could easily be missed in the far wake [21].
All the former work described here does not take into account the unsteady portion of
the flow presumably generated by the wing’s motion. Rayner et al. [22] performed
measurements on starlings in undulating flight in a wind tunnel and showed that the geometry
of the flight path depends upon wingbeat kinematics, and that neither the flapping nor the
gliding phases of flight occur at constant speed or at constant angle to the horizontal. The bird
gains both kinetic and potential energy during the flapping phases making it difficult to
model. Rayner et al. [22] indicate that such speed variation can provide significant savings in
mechanical power in both bounding and undulating flight. Recently, high-speed PIV systems
have become available for animal flight research with which the wake structure can be
analyzed at a high temporal resolution [23, 24]. In these studies, the wake is sampled using
PIV images taken at a typical frequency of 200 Hz in a transverse plane (vertical spanwise)
referred to as the Trefftz plane [25]. The three-dimensional wake is assembled by identifying
coherent streamwise structures, such as the tip vortex [26], and, in consequence, the time
varying flight forces have been estimated based on this method [27-30]. However, the focus
in such studies has been on the lift force and not on the relation between drag and thrust. It is
also noteworthy that PIV measurements in the Trefftz plane consist of substantial uncertainty
in the estimated velocity and its gradients due to the set-up complexity and the nature of the
PIV technique when performing flow measurements with a strong out of plane velocity
component [25]. Therefore, conclusions drawn based on these measurements should be
carefully utilized. To date, there is no available volumetric technique capable of performing
full three-dimensional measurements of high speed flows in air. The existing techniques such
as StereoPIV provide three velocity components but not gradients. In addition, the accuracy
level of this technique in reconstructing the third dimension is not high [31, 32]. Therefore,
the current models and quantitative estimations of forces behind the wake in the Trefftz plane
are subject to relatively large error [25].
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Currently, most of the aerodynamic models for birds are based on fixed wings in steady
flow [33]. While the quasi-steady values seem to be valid in many bird wakes, it is far from
proof that the approach is valid. Therefore, the current study addresses the near wake
variations behind a freely flying bird in time and space with a particular focus on the unsteady
aerodynamics of the flow that results from the flapping wing. The change of velocity with
time is the key parameter that marks the unsteady effect and its variation is examined in terms
of the drag and thrust balance.
2 Experimental setup
2.1 Wind tunnel
The experiments were performed in the closed-loop hypobaric climatic wind tunnel at
the Advanced Facility for Avian Research (AFAR) at the University of Western Ontario. The
test section of this wind tunnel has a cross-sectional area of 1.2 m2, is preceded by a 2.5:1
contraction, and is enclosed in a hypobaric chamber. The width, height and length of the test
section are 1 m, 1.5 m, and 2 m, respectively. An open jet exists between the downstream end
of the test section and the diffuser for the purpose of introducing the live bird into the wind
tunnel during the experiments. The turbulence intensity is lower than 0.3% at the location
where the measurements were taken. A fine net was placed at the upstream end of the test
section to prevent the bird from entering the contraction, which was not observed to alter the
turbulence significantly. The flight conditions were at atmospheric pressure, a temperature of
15 °C, and relative humidity of 80%.
2.2 The Bird - European Starling
The wake measurements (as illustrated in Figure 1) were taken from a European starling
that had been trained to fly in the AFAR wind tunnel. The bird's wings had an average chord,
c, of 6 cm, a maximum wingspan of b=38.2 cm and an aspect ratio (wingspan squared divided
by the wings lifting area), AR, of 6.4. A typical cruising speed of U∞=12 m/s was chosen for
the experiments based on the comfort of the starling and its ability to fly for prolonged
periods of time during the testing. The wingbeat frequency, f, was 13.3 Hz on average, and the
average peak-to-peak wingtip vertical amplitude, A, was 28 cm. These quantities correspond
to a chord-based Reynolds number of 4.8·104, a Strouhal number, St=Af/U∞=0.30, and a
reduced frequency, k=πfc/U∞=0.20. At the time the experiments were performed the bird had
a mass of 78 g and a lateral body width of 4 cm.
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Due to the powerful laser operating within a few chord lengths of the bird’s tail, two
precautions were taken to ensure the bird’s safety. Goggles made of a flexible, optically
dense, polymer material (Yamamoto Cogaku Co YL 600) were designed to protect the bird’s
vision as well as to reduce the potential of the light sheet frightening the bird. After an
accommodation period of 20 minutes in a cage, the bird would fly normally in the wind
tunnel while wearing the goggles. In addition, for preventing direct contact between the bird
and the light sheet, a collection of optoisolators operated by six infrared transceivers were
integrated into the PIV system. The function of the optoisolators was to trigger the laser only
when the bird was in a desired position upstream of the PIV field of view, thus ensuring that
the bird was in a position where it was not in danger of being hit by the laser. All animal care
and procedures were approved by the University of Western Ontario Animal Use Sub-
Committee (protocols 2006-011, 2010-216).
2.3 Flow velocity and kinematic measurements
Flow measurements were taken using the long-duration time-resolved PIV system
developed by Taylor et al. [34]. Olive oil particles, 1µm in size [35] were introduced into the
wind tunnel using a Laskin nozzle from the downstream end of the test section so that it did
not cause a disturbance to the flow in the test section or to the bird. The PIV system consists
of an 80 W double-head, diode-pumped, Q-switched, Nd:YLF laser at a wavelength of
527 nm and two CMOS cameras (Photron FASTCAM-1024PCI) with spatial resolution of
1024x1024 pixel2 at a rate of 1000 Hz. The PIV system is capable of acquiring image pairs at
500 Hz using the two cameras for 20 minutes continuously. In the current experiments, one
camera was used to record the bird kinematics during the wingbeat and the other was used for
PIV measurements in the wake. The PIV camera’s field of view was approximately 12 x 12
cm2 in size, or 2c by 2c. Vector fields were computed by OpenPIV [34] using 32 x 32 pixel2
interrogation windows with 50% overlap, giving a spatial resolution of 32 vectors per chord.
In the current experiments, 4 600 vector maps were recorded, and out of this dataset, 650
vector maps contained features of the near wake behind the starling's wing. The PIV data
were measured 4 wing chord lengths (~0.24 m) behind the right wing, and therefore it took 20
ms for events generated at the wing to enter the PIV field of view. The wake was sampled in
the parasagittal plane (9 x 9 cm2) at 2 ms intervals (500 Hz), so that both the downstroke and
upstroke phases were temporally resolved.
The streamwise and vertical positions of the bird for all measurements were recorded
simultaneously with the flow field measurements. The field of view in these recordings had
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an area of 9c by 9c. Figure 2 depicts a sample image of the starling flying in the tunnel as
captured by the camera. The box marked with “PIV” indicates the location of the measured
velocity fields from the PIV system. In addition, a floor-mounted camera operating at 60 Hz
was used to record the spanwise position of the bird as well as the laser sheet illumination.
These images allowed for the identification of the measured PIV plane in respect to the
position of the wing; therefore, the wake velocity field could be associated with the spanwise
location across the wing or the body. The floor-mounted camera was not synchronized with
the PIV system; therefore, the two time histories were synchronized manually based on the
presence of light from the laser firing in the images. Once synchronized, spanwise positions
were assigned to the wake data captured at 500 Hz based on interpolation from the
simultaneously recorded spanwise positions recorded at 60 Hz.
An error analysis based on the root sum of squares method has been applied to the
velocity data and the wing kinematics. The errors were estimated as: 2.5% for the
instantaneous velocity values, 12% for the instantaneous vorticity and 3% for the drag values
[36]. The error introduced in the kinematic analysis resulted from the spatial resolution of the
image and the lens distortion leading to an estimated error of 5% in the wing displacements.
3 Results
A number of wake velocity maps were sampled where the starling was flying in a
steady flapping mode without performing any maneuvers. The data discussed herein was
selected from a broad acquisition batch where the bird was flying continuously for a few
minutes (see Figure 3, where the streamwise velocity at the wake is depicted as a time series).
The selection criterion was based on the flight mode chosen: no net acceleration of the bird
over a wingbeat cycle, as observed from the high speed imaging. The wing kinematics and the
flow analysis are presented in the following sections. The analysis includes four sets of
wingbeats each comprising a downstroke phase and an upstroke phase. The first three
wingbeat sets contain a total of 110 velocity maps and kinematic images acquired
simultaneously. These sets feature three consecutive wingbeats (referred to as wingbeats 1, 2
and 3, according to their order of appearance). A fourth wingbeat set (wingbeat 4) contains 43
vector maps and kinematic images.
3.1 Kinematic Analysis
Figure 4 illustrates the starling in different positions during wingbeats 1, 2, 3 and 4.
Since the purpose of this study is to estimate the streamwise forces acting on the bird, it is
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imperative that the bird accelerates negligibly in this direction. From the kinematics shown in
Figure 4 it is observed that, during all four wingbeats, the starling does not accelerate
noticeably in the streamwise direction. Thus, analysis of the wake aerodynamics during these
wingbeats can be performed assuming negligible acceleration of the bird in the streamwise
direction.
In order to characterize the kinematics of the bird, several parameters have been
estimated which are commonly used to evaluate bird flight characteristics beginning with the
flapping frequency and Strouhal number [2] (see Table 1). The flapping frequency was
calculated according to the inverse of the period, T, of each wingbeat f = 1/T, and the Strouhal
number according to
St = fAU!
. (1)
where A is the amplitude of the wingtip and U∞ is the free stream velocity. Taylor et al. [37]
suggest that, for a wide variety of animals (including fish, birds and insects) efficient cruising
locomotion requires that the Strouhal number be in the range of 0.2 < St < 0.4, and for birds
during cruising flight it should be nearly 0.2. The current results fall within this predicted
range, and the minimum value (St = 0.24) approaches the value for cruising flight during
wingbeat 4.
The wingtip angle of attack, φ, is computed by assuming that the wing moves up and
down through its amplitude at a constant vertical speed
v = 2 fA . (2)
For a wing moving forwards with a streamwise velocity U∞, the wingtip will move either up
or down with a maximum ‘zigzag’ angle given by [38]:
tan(! ) = 2 fAU!
(3)
Substituting the Strouhal number from Eq. (1) yields:
! = tan!1(2St) . (4)
For a bird flapping its wings up and down in the vertical plane and keeping the wing
chord horizontal all the time (e.g., Figure 4), the wingtip angle of attack can be approximated
as φ on the downstroke and - φ on the upstroke [38]. It is observed that the minimum mean
wingtip angle of attack during the downstroke phase (φ=26º) occurs for wingbeat 4 and that
the value is larger than the stalling angle of conventional fixed wing aircraft of approximately
15o [39]. However, as shown by Nachtigall and Wieser [40], the angle of attack varies from
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zero at the shoulder to a maximum value at the wingtip. Thus, on average, the attack angle
over much of the wingspan is lower than the stalling angle of 15º.
3.2 Wake characteristics
In the previous section, the wing kinematics were quantified for each of the four
complete wingbeats. In this section, the wake is characterized in terms of aerodynamic forces
and vorticity content. The vorticity in the wake is computed directly from the PIV data using
a least squares differentiation scheme [36]. To determine if the vorticity as measured at the
near wake is sufficient for the force estimations, the peak vorticity in the current data behind
the starling is compared with former works [1, 7, 18, 20, 41], as depicted in Figure 5. The
peak spanwise vorticity measured in the wakes of several flapping wing animals is displayed
in Figure 5 for the purpose of contextualizing the current measurements in the starling wake
among other flapping wing animal studies. In Figure 5, the spanwise vorticity is normalized
by the mean chord and wind speed of each respective study. Since peak vorticity measured in
the wake of a cruising animal varies gradually over the range of flight speeds [18], peak
values of spanwise vorticity are included for both extremes of the natural speed range where
possible.
It is observed that the peak normalized vorticity (4.1) as depicted in Figure 5 of the
starling in the present study is larger than values of peak vorticity from animals in cruising or
fast flight (red and purple bars). Peak values of vorticity from birds and bats flying at the
lower end of their natural speed range (blue and green bars) are more comparable to what is
displayed by the starling.
3.2.1 Wake Reconstruction
During flapping flight, bird wings change position causing the momentum and
circulation in the wake to vary. In many simplified models, the wake changes in a periodic
manner where the downstroke and upstroke phases have different signatures [12, 18, 43]. In
order to characterize the effect of the flapping action on the near wake behind the starling and
its impact on the aerodynamic performance, sequences of velocity maps have been
reconstructed. This procedure was performed using PIV data collected at a sampling rate of
500 Hz – significantly higher than the 13.3 Hz wingbeat frequency. Therefore, a pattern of
vorticity appearing in one frame also appears in the consecutive frame; only phase-shifted.
The wake composite is formed by plotting sequential instantaneous vorticity fields computed
from PIV data and by matching patterns in the vorticity fields with a shift. The offset of the
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nth successive PIV images is calculated as Uc·Δt ·n. The convection velocity, Uc, is the
velocity at which the characteristics of the wake collectively travel downstream. In the present
study, wake composites have been generated using the free-stream velocity (U∞) as a
convection velocity. The generation of a wake composite provides a useful visualization tool
for observation of the wake dynamics over the time series of a wing beat cycle. What appears
as “downstream” in the wake composite happens earlier, while what appears “upstream” in
the composite happens later meaning that the generation of the wake composite invokes
Taylor’s hypothesis in which the characteristics of the flow are a frozen spatial pattern
advected through the field of view.
Figure 6 demonstrates the wake reconstruction procedure: initially two consecutive
spanwise vorticity fields, along with the velocity fluctuations, are put side by side.
Afterwards, a vorticity pattern classification process is performed on each image, which
eventually assists in the identification of similar patterns between the two images according to
similarity in size, shape, direction and value. Figure 6 shows four different negatively-signed
(A-D) patterns and two different positively-signed (E and F) patterns. It can be seen that the
different patterns of vorticity move downstream during the time difference between the
consecutive images (2 msec).
3.2.2 Wake Evolution
The wake features are presented through fluctuating velocity and vorticity fields as
depicted in Figure 7. The set of figures describes the wake of the freely flying starling during
the four different wingbeats (as defined in §3.1). Each wake pattern consists of 24 consecutive
fields displaying the spanwise vorticity, ωz(x, y), varying from -650 to 650 sec-1. In addition,
the spatially averaged velocity has been subtracted from each frame, so the velocity vectors
displayed are fluctuations.
Using the floor-mounted camera, the wake patterns presented in Figure 7 were
determined to be captured in a plane that is, on average, approximately 2.5 cm from the right
wing root (14% of the wing length). Figures 8 and 9 demonstrate the different wing sections
being intersected with the laser sheet as a consequence of the starling's small spanwise
movements throughout the wingbeats. It should be noted that, in the four wingbeats presented
in this study, the starling was recorded with the minimum possible spanwise, vertical and
streamwise movements.
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The most immediate observation from the wake reconstructions in Figure 7 is the
periodicity of the wake over the shedding cycle. In light of the topology of spanwise vorticity,
it is convenient to discuss the wake in terms of a top half and a bottom half where the wake
center is defined by the location of greatest velocity deficit. As would be expected, the top
half of the wake is composed primarily of negative spanwise vorticity and the bottom half of
the wake by positive spanwise vorticity. It is also noted that the quantity of lift producing
vorticity (i.e., negative) is greater than that of positive vorticity.
In the available literature, there has been considerable focus on the vortex topology in
bird wakes [18, 20, 28]. A comparison with these works shows that there are qualitative
similarities in the vorticity structure between the current measurements close to the body
(along the span) and those shown by Henningsson et al. [7] for a swift. From their
measurements of the wake at 10 chord lengths downstream of a flying swift, Henningsson et
al. [7] suggest that the tip vortices for the swift are connected by spanwise vortices. The
measurement plane in the current study is significantly closer to the bird in the streamwise
direction (4 chord lengths) offering new perspective on wake development. Distinguishing
coherent vortices from shear in a real fluid flow is not trivial [42-44]. Many wakes are
typified by high shear and bird wakes are no different. Vorticity alone has shown to be
inadequate in distinguishing a vortex from an area of high shear [43], and the PIV data in the
current study are of insufficient resolution to distinguish if the majority of the vorticity
observed in Figure 7 is due to coherent vortices or high shear. Thus, if there are spanwise
connecting vortices, they should be relatively small at the location of our measurement plane.
The strong vorticity observed in the wake implies the presence of shear and, as a result, drag.
In the next section, the sectional drag force is examined quantitatively.
3.3 Drag estimates
Consider a section of the bird wing as a two-dimensional body in an incompressible
flow as sketched in Figure 10. The body is located within a control volume (abcsdefghia) with
its width in the z direction being unity. Inside the control volume, the integral form of the
momentum equation is [45]
')( RpdSudSudVut abhiSV
−−=⋅+∂
∂∫∫∫∫∫∫∫
ρρ
(5)
where u is the velocity, ρ is the density (constant), p is the pressure and R' is the resultant
aerodynamic force per unit span exerted on the body by the normal and shear stresses acting
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at the body surface. Note that the viscous terms have been ignored as they scale with Re-1.
The integrals are taken over the control volume, V, enclosed by the surface, S. Using the x-
component of Eq. (5), the aerodynamic drag per unit span, D', is
∫∫∫∫∫∫∫ −⋅−∂
∂−=
abhix
SV
pdSudSuudVt
D )()(' ρρ
.
(6)
A positive value of D' is defined as drag and a negative one as thrust. For simplicity, the
resultant force, D', is referred to as drag per unit span where negative drag per unit span refers
to thrust. For S, estimated sufficiently far from the body where the pressure is assumed
constant, and equal to the undisturbed free-stream pressure p∞, Eq. (6) becomes:
∫∫∫∫∫ ⋅−∂
∂−=
SV
udSuudVt
D )(' ρρ
.
(7)
By definition, u is parallel to the streamlines and dS is perpendicular to the control
surface. Thus, for streamlines ab, hi and def, the multiplication 0=⋅ dSu . In addition, the
planes cd and fg are adjacent to each other, so their contribution to the second term in Eq. (7)
cancel each other. As a result, the second term in Eq. (7) consists of contributions only from
sections ai and bh (where dydS = ). Therefore, Eq. (7) is expressed as
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−
∂
∂−= ∫∫∫∫
b
h
a
iS
dyudyuudxdyt
Dabhi
22
21' ρρρ
.
(8)
Using the integral form of the continuity equation and multiplying by u1 (a constant in the
current case),
∫∫ =b
h
a
i
dyuudyu 1221 ρρ
(9)
and substituting Eq. (9) into Eq. (8) leads to
∫∫∫ −+∂
∂−=
b
hS
dyuuuudxdyt
Dabhi
)(' 212ρρ.
(10)
Therefore, the drag is composed of two terms: steady (second term) and unsteady (first term).
The steady drag per unit span, referred to as the velocity deficit drag in classical
aerodynamics, can be derived from the second term in Eq. (10) and expressed as
∫ −= ∞
h
Steady dyuUuD0
)(' ρ
(11)
where h is the wake vertical extent of the PIV velocity field. The so-called ‘unsteady drag’ per
unit span can be derived using the first term in Eq. (10) and expressed as
13
∫ ∫∂∂
−≈h l
Unsteady udxdyt
D0 0
' ρ
(12)
where l is the streamwise extent of the PIV velocity field.
It should be noted that the full area integral (or, volume integral per unit span) cannot be
directly computed as it appears in Eq. (10) since only a portion of the control surface
enveloping the bird wing is measured in the current experiments (e.g., Figure 2). However,
Eq. (12) is used as an approximation to the entire area integral bound by the control surface
shown in Figure 10. Figures 11 and 12 describe the time variation of the steady and unsteady
drag per unit span as computed from Eq. (11) and (12) for the four different wingbeats (see a-
d in Figures 11 and 12) as depicted in Figure 7. The integrals were performed for each
instantaneous velocity field of the starling’s wake. For the steady drag per unit span, different
streamwise velocity profiles were sampled at different x-positions for each velocity field map.
Subsequently, these profiles within one PIV vector map were spatially averaged into one
profile describing the velocity deficit (steady drag per unit span) similar to the procedure
described in [18]. This procedure is similar to a spatial windowing average in order to smooth
out some of the variations within each vector map, and it is noted that the general trend over
the wingbeat cycles does not change using this procedure. Each point in Figures 11 and 12
represents the integral value depicted from each velocity field yielding a time evolution of the
drag in the near wake. Figures 11e and 12e depict the averaged drag profiles for the four
wingbeats. The uncertainty of the computed drag values in Figures 11 and 12, estimated in
§2.3, is similar to the size of the markers. Notable differences are observed between the
steady and unsteady components of the horizontal momentum, shown in Figures 11 and 12.
Figure 11e shows that the averaged steady drag is positive over the entire wingbeat cycle
except for a short period in the transition from downstroke to upstroke. Contrary to the steady
portion, the unsteady contribution to the drag as depicted in Figure 12e is negative during
both the downstroke and upstroke while positive during the transition phase. The steady drag
values reach 1 and 0.5 N/m during the upstroke and downstroke, respectfully. The negative
values as calculated for the unsteady portion reach a minimum value of -0.5 N/m. These
differences are discussed in detail in the following section.
4 Discussion
In classical aerodynamics, the profile drag (Eq. (11)) inherently assumes that ∂u/∂t = 0
everywhere in the chosen control volume. While this assumption is reasonable in the wake of
a section model mounted in a wind tunnel, it seems unlikely that this condition is satisfied in
14
the wake of a freely flying bird. However, experimental measurements of the entire control
volume remain prohibitive and many studies have approximated the drag force through the
profile drag [18, 46]. In the current study, the profile drag has been estimated in the same
manner over four complete wingbeats of a freely flying starling (Figure 11). The steady drag
was demonstrated to vary significantly during the wingbeat cycle yielding higher profile drag
during the upstroke. This variation in the drag force over the wingbeat cycle is consistent with
earlier studies [47], which suggested that the upstroke phase generates more drag than the
downstroke phase.
The profile drag developed over the wing is manifested through the wake velocity
profile. In the case of flapping wings (or in any event where the flow is disturbed by some
external motion), the velocity field changes spatially and temporally. Since the bird is flying
freely, and the kinematic images (Figure 4) demonstrate that the bird does not accelerate in
the streamwise direction, there is no net momentum change in the streamwise direction.
Since the profile drag is the term relevant to the overall streamwise momentum balance of Eq.
(5) then one would expect a momentumless velocity profile (see Figure 13). However, as
indicated by the profile drag, which is generally positive (Figure 11e), as well as in profiles
presented in previous studies [18], it appears that another force is required to balance the
streamwise momentum.
From the momentum balance (Eq. (5)) it is expected that the compensating force for the
profile drag is the volume integral containing the ∂u/∂t term. Considering the control volume
drawn in Figure 10, it is plausible to assume that ∂u/∂t = 0 everywhere at the flow upstream of
the wing since the wind tunnel is operating at a constant wind speed. Conversely, ∂u/∂t is not
expected to be zero around the wings and especially in the wake. It is expected that ∂u/∂t is
greater than zero in the wake during the downstroke since work (e.g., the flapping motion) is
an input to the wake [45]. However, during the upstroke it remains unclear if this term is
negative or positive. The estimate of the ‘unsteady drag’ (Eq. (12); Figure 12) uses a portion
of the volume as an approximation to the total volume integral of ∂u/∂t. The results presented
in Figure 12e demonstrate that ∂u/∂t is smaller than zero as expected during the downstroke as
the bird generates lift and propels itself forward [45]. At the beginning of the upstroke, the
volume integral of ∂u/∂t appears to be positive; however, once the steady drag reach a
maximum value halfway through the upstroke (Figure 11e), the unsteady drag once more
becomes negative indicating that ∂u/∂t > 0 as it was during the downstroke.
It should be noted that performing PIV measurements around freely flying birds limits
the capability of capturing the entire span of the wing; therefore, our estimates are based on a
15
sectional measurement. However, the current dataset offers much lower uncertainty of the
streamwise velocity compared to attempts to capture the entire volume using the Trefftz plane
[25]. Drag variations may also arise due to the wing flexing, which acts to minimize the drag
during the second half of the upstroke phase [2]. In addition, it is well known that a
significant amount of thrust is generated at the outer part of the wing [2]. However, the time-
resolved streamwise velocity measurements in the near wake of a freely flying starling
suggest that for a complete understanding of the drag and thrust relationship in bird wakes the
importance of ∂u/∂t cannot be neglected. Furthermore, the flow mechanisms underlying the
present measurements of ∂u/∂t are expected to correlate to the bird’s use of unsteady
aerodynamics and improved efficiency.
Many studies that investigate flying animals utilize quasi-steady models to analyze the
flapping mechanism [16, 18, 45]. Quasi-steady analysis of flapping flight generally assumes
that the aerodynamic forces in flapping flight can be composed from the various
instantaneous wing configurations, as they would behave in an equivalent series of steady
flows. However, our estimation of the steady drag force shows that it is mostly positive
indicating that the bird should be decelerating if acted on by this force alone. The kinematic
observations show that the bird is not accelerating in the streamwise direction, so there should
be a balancing force that is not accounted for in the steady aerodynamics. The approximation
of the unsteady contribution to the streamwise force indicates that the balancing force to the
steady velocity deficit drag is most likely due to unsteady aerodynamics, thus revealing an
inadequacy of quasi-steady approaches. Future studies measuring a greater spanwise
distribution of drag forces are necessary to obtain the complete description of steady versus
unsteady aerodynamics.
5 Conclusions
In this study, a long-duration time-resolved PIV system was used to obtain accurate,
time-resolved measurements of the streamwise velocity in the wake of a freely flying
European Starling flying in flapping flight at the AFAR hypobaric wind tunnel. The system is
capable of capturing images for 20 min continuously in order to characterize unsteady
phenomena within a given flow field. A total of 4,600 vector maps were analyzed in the
current study, and four wingbeat cycles were identified within this data set. The identification
was performed by using an additional high-speed camera that recorded flight kinematics and
was synchronized with the PIV. The kinematic analysis showed that during the four
16
wingbeats used to analyze the wake, the bird did not accelerate or decelerate significantly in
the streamwise direction.
The wake topology of the starling was characterized using a wake reconstruction based
on patterns from the instantaneous vorticity fields where the measurement plane of the
velocity was close to the wing root of the bird. The resolution of the data were insufficient to
determine if spanwise vortices were present in the near wake or if the observed vorticity was
due to the shear created by the flow over the bird’s wings. Thus, any connecting spanwise
vortices that exist at this stage of the wake development should be relatively small.
The time variation of the profile drag per unit span from classical aerodynamics over
each of the four different wingbeat cycles was approximated using the PIV data. Inherent in
the calculation of profile drag is the assumption that ∂u/∂t is zero everywhere or integrates to
zero instantaneously, yet this does not necessarily correspond to the case for flapping flight.
As with previous studies, the integration of the velocity profiles over the measurement plane
admittedly misses the remaining span of the wake, yet clear trends have been observed. It was
found that the profile drag term was almost always positive; however, the bird was not
observed to noticeably decelerate. Thus, there should be a compensating force to this classical
drag term. It was observed that ∂u/∂t is generally negative during the downstroke from the
current dataset; however, the results also show that during the upstroke ∂u/∂t is generally
positive. The approximation of the unsteady term suggests that unsteady aerodynamics may
provide some thrust in the overall streamwise force balance.
The role of the unsteady portion of the flow on the flight efficiency of birds is yet to be
determined and still remains an open question. Yet, the current results shed light on the role of
the unsteadiness during flight and its impact on drag/thrust. In addition, future studies are
required to assess how the spanwise variations of these forces affect the balance between drag
and thrust.
Acknowledgements
Z.J. Taylor gratefully acknowledges the support of the Tel Aviv University Post-doctoral
Fellow Scholarship. R. Gurka and C.G. Guglielmo gratefully acknowledge funding from the
NSERC Discovery Grants Program, and the Canada Foundation for Innovation and Ontario
Research Fund for construction of the AFAR at the University of Western Ontario.
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17
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20
Figure 1 Illustrative scheme of the experimental setup system.
Figure 2 The large image shows the kinematic camera field of view and the small window marked “PIV” is the PIV camera field of view.
High speed camerafor the starling
High speed camerafor the wake
Double-pulsed laser
Laser light sheet
Wind tunnel
Wake field of view
Starling
A=1.2m2
P=1atmT=15oCφ=80%
2pixel1024x1024: CamerasCMOS10bit , 1000Hz
-Nd:YLF, double:Laserhead diode-pumped laser Wavelength of 527nm80W at 3000Hz
Olive oilParticles:d = 1µm
Air + Particles
Innovative Long-Duration Time-Resolved PIV System
12 m/s=U∞
3 m
Computer and Synchronizer
60cm
Starling
Wake Measurements
PIV
60cm
12cm
12cm24cm
21
Figure 3 Variation of the average streamwise velocity in the wake with time.
Figure 4 Different positions of the starling during the four wingbeats. The photographs are labeled so that the number corresponds to the wingbeat number, ‘a’ marks the beginning of the downstroke, and ‘b’ marks the beginning of the upstroke.
4 Wingbeats
10.5
11
11.5
12
12.5
13
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5
U
[m/sec]
Time [sec]
(1a) (1b) (2a) (2b)
(3a) (3b) (4a) (4b)
22
Figure 5 A comparison of the peak spanwise vorticity measured in the wake of flapping animals from several studies. Superscripts refer to the work of: (1) [18]; (2) [20]; (3) [41]; (4) [1]; and (5) [7]. Abbreviations used in the figure represent the thrush nightingale (TN), house-martin (HM), and Pallas’ long-tongued bat (PLTB). Measurements from the Pallas’ long-tongued bat come from the inner wing (z/bsemi<0.4) and outer wing (z/bsemi>0.75).
Figure 6 Two consecutive spanwise vorticity fields (t2= t1+2 msec). The air flows from left to right and each frame size is 9 x 9 cm2.
+ +-
---
++
-
--
-
A
A
BB
C C
DD
EE
FF
[sec-1]
t1 t2
ωz [sec-1]
23
Figure 7 Reconstruction of the starling's wake consisting of four wingbeats as though the starling flew from right to left. The average spatial flow has been subtracted, thus the vectors displayed are the velocity fluctuations. The contours represent the spanwise vorticity in each wingbeat and the half-wavelengths for the downstroke (λd) and the upstroke (λu) are noted for each wingbeat. Wingbeat numbers go from top to bottom: (a) 1, (b) 2, (c) 3, and (d) 4.
Figure 8 The location of the laser sheet with respect to the right wing during wingbeats 1, 2, and 3.
λu=0.50m λd=0.43m
λu=0.43m λd=0.38m
λu=0.43m λd=0.43m
λu=0.58m λd=0.43m
ωz [sec-1]
(a)
(b)
(c)
(d)
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.05 0.10 0.15 0.20 0.25
Distancefrom Shoulder/Semi-Span
Time [sec]
Wingbeat 1
Wingbeat 2
Wingbeat 3
24
Figure 9 The location of the laser sheet with respect to the right wing during wingbeat 4.
Figure 10 Control volume around a two-dimensional body in a uniform free stream.
Fig. 11a Steady drag per unit span versus time, as computed according to Eq. (13), for wingbeat no. 1.
0.00
0.20
0.40
0.60
0.80
1.00
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Distancefrom Shoulder/Semi-Span
Time [sec]
y
x
u1=const.=U∞
u2(y)
2D body
a
b
h
i
ed
f
c
g
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
D'Steady
[N/m]
Time [sec]
Downstroke Upstroke
25
Fig. 11b Steady drag per unit span versus time, as computed according to Eq. (13), for wingbeat no. 2.
Fig. 11c Steady drag per unit span versus time, as computed according to Eq. (13), for wingbeat no. 3.
Fig. 11d Steady drag per unit span versus time, as computed according to Eq. (13), for wingbeat no. 4.
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
D'Steady
[N/m]
Time [sec]
Downstroke Upstroke
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
D'Steady
[N/m]
Time [sec]
Downstroke Upstroke
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
D'Steady
[N/m]
Time [sec]
Downstroke Upstroke
26
Fig. 11e Averaged steady drag per unit span versus time, as computed according to Eq. (13).
Fig. 12a Unsteady drag per unit span versus time, as computed according to Eq. (14), for wingbeat no. 1.
Fig. 12b Unsteady drag per unit span versus time, as computed according to Eq. (14), for wingbeat no. 2.
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
D'Steady
[N/m]
Time [sec]
Downstroke Upstroke
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
D'Unsteady
[N/m]
Time [sec]
Downstroke Upstroke
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
D'Unsteady
[N/m]
Time [sec]
Downstroke Upstroke
27
Fig. 12c Unsteady drag per unit span versus time, as computed according to Eq. (14), for wingbeat no. 3.
Fig. 12d Unsteady drag per unit span versus time, as computed according to Eq. (14), for wingbeat no. 4.
Fig. 12e Averaged unsteady drag per unit span versus time, as computed according to Eq. (14).
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
D'Unsteady
[N/m]
Time [sec]
Downstroke Upstroke
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
D'Unsteady
[N/m]
Time [sec]
Downstroke Upstroke
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
D'Unsteady
[N/m]
Time [sec]
Downstroke Upstroke
28
Figure 13 Schematic examples of a drag wake (a), momentumless wake (b) and a jet wake (c).
Table 1 The wingbeat frequency, Strouhal number and the wingtip angle of attack for the four wingbeat cycles
Parameter Wingbeat
Average* 1 2 3 4
f ][Hz 12.8 14.7 13.9 11.9 13.3
St 0.27 0.34 0.33 0.24 0.30
φ ][o 29 34 33 26 31
* The average value of each parameter for the all wingbeats
U∞ U∞ U∞
(a) (b) (c)
29